Flows on quaternionic-Kaehler and very special real manifolds

TL;DR

This paper studies gradient flows on quaternionic-Kaehler and very special real manifolds related to 5D supergravity, analyzing the Hessian of an energy function at critical points and revealing properties depending on manifold symmetry.

Contribution

It provides new calculations of the Hessian of the energy function on quaternionic-Kaehler manifolds and characterizes critical points for symmetric and non-symmetric cases.

Findings

Existence of Killing vector fields with split signature Hessian at critical points.

Symmetric quaternionic-Kaehler manifolds have non-degenerate local extrema.

Non-symmetric homogeneous cases exhibit degenerate local minima.

Abstract

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient flows are generated by the `energy function' f = P^2, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kaehler manifolds. For the homogeneous quaternionic-Kaehler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p in M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kaehler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kaehler manifolds we find degenerate local minima.